Multistep DBT and regular rational extensions of the isotonic oscillator

TL;DR

This paper extends a systematic method for generating solvable rational extensions of shape invariant potentials, applying it to the isotonic oscillator to produce new isospectral potentials linked to exceptional Laguerre polynomials.

Contribution

It introduces a multistep scheme for rational extensions of shape invariant potentials, specifically applied to the isotonic oscillator, revealing new isospectral potentials with explicit eigenstates.

Findings

New towers of regular rational extensions of the isotonic oscillator

Extensions are strictly isospectral and inherit shape invariance

Eigenstates involve exceptional Laguerre polynomials

Abstract

In some recent articles we developed a new systematic approach to generate solvable rational extensions of primary translationally shape invariant potentials. In this generalized SUSY QM partnership, the DBT are built on the excited states Riccati-Schr\"odinger (RS) functions regularized via specific discrete symmetries of the considered potential. In the present paper, we prove that this scheme can be extended in a multistep formulation. Applying this scheme to the isotonic oscillator, we obtain new towers of regular rational extensions of this potential which are strictly isospectral to it. We give explicit expressions for their eigenstates which are associated to the recently discovered exceptional Laguerre polynomials and show explicitely that these extensions inherit of the shape invariance properties of the original potential.